Lagrangian Systems in the Presence of Singularities

Abstract

ABSTRACT. In this paper we study dynamical systems embedded in a conservative field of forces, whose potential is "singular." We look for T-periodic solutions of these systems by variational methods. Introduction.In this paper we look for T-periodic solutions of the Lagrangian system of ordinary differential equations:where the Lagrangian function C(t,q, £) is given, as usual, byÍGR, q,£eRN, and dij(t,q), bi(t,q), c(t,q), V(t,q) are C1 real-valued functions, T-periodic in t. Moreover we suppose that the "potential" V(t,q) is defined in R x fi, where fi is an open subset of Rw, and V{t,q) -► -oo as?-» dQ. Many authors have studied this problem in the case when fi = R^ (so dQ = 0) under various assumptions on the growth of V(t,q) as \q\ -► oo: cf., for instance, [2, 3, 5, 9, 10]. W. B. Gordon was the first to study our case by means of variational methods, and we refer to [6,7] for the physical motivation of the problem (cf. also the end of this section). Finally we refer to [1,8] for the case V(t,q) -► +oo as q^dQ {Ü¿RN).In this paper we suppose that:

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

“…The main difficulty of avoiding collision orbits (for which (2) is not satisfied) was overcome by introducing the so-called "strong force" assumption. His result, later improved by Capozzi, Greco and Salvatore [10], can be stated for our convenience as follows. The proof, originally given only for period T , consists in minimizing the associated functional over the orbits which turn around the origin exactly once in their period time.…”

Periodic motions in a gravitational central field with a rotating external force

“…The use of topological methods in the study of singular systems goes back to Poincaré [26]. More recently, variational methods have been successfully employed in, e.g., [1,4,5,7,9,12,20,27]. See also the book [2], and the references therein.…”

Periodic solutions of singular radially symmetric systems with superlinear growth

“…First of all, by (H1) and (H3), there is a α > 1 such that g(r ) ≤ −1 for every r ∈ 0, 1 α , and g(r ) ≥ 1 for every r ≥ α. We can now easily construct a functionĝ(r ), which coincides with g(r ) on 0, 1 α ∪ [α, +∞[, and satisfies (8). Settinĝ e(t, r ) = e(t, r ) +ĝ(r ) − g(r ), we have thatĝ andê still verify all the assumptions of Theorem 1.…”

“…After these modifications, by (8), the point 1 is a minimum point for G. We can assume, defining G as in (3), withr = 1, that G(1) = 0. We may write the solutions of (1) in polar coordinates:…”

“…Starting with Gordon [25], many attempts have been done in order to deal with periodically perturbed systems of this type, even without radial symmetry. See, e.g., [1,8,12,37] and the references therein, for the variational approach, and [9,26,29,39], where topological methods were employed, instead. The repulsive case, modelling the Coulomb equation governing the motion of electrical charges, has been treated, e.g., in [10,16,26,38].…”